Entropy matters

More is different

Philip Anderson (Science 1972)

Complexity emerges when many simple units interact: studying the individual components does not explain the emergence of collective properties

Beyond the usual phases of matter

We sometimes oversimplify…

Extreme examples of phases of matter

Beyond the usual phases of matter

We sometimes oversimplify…

Complex fluids and soft condensed matter

• Soft condensed matter includes assembllies of colloids, polymers, surfactants, and biological macromolecules and much more.

  • Term due to Pierre-Gilles de Gennes (Nobel 1991)

• Often, many ofthese are also referred to as complex fluids.

• These materials are easily deformed and show complex, disordered structures.

• Their properties are determined by a balance between energy and entropy.

• Thermal fluctuations play a major role in their behavior.

• Understanding them requires statistical mechanics .

Thermal fluctuations and entropy

• Helmholtz Free energy per particle f = u-Ts

• Entropy approximately counts the number of arrangements per particle s = \dfrac{k_B}{N} \ln{\Omega}. For hundreds of arrangements per particle one has s=O(1) k_B

• Fluctuations of the internal energy are on the same scale as thermal fluctuations: \boxed{\Delta u \sim k_B T} where k_B is the Boltzmann constant and \Delta u indicates standard deviations from the average internal energy.

Thermal fluctuations: single particle in a doyble well

viewof params = Inputs.form({
    T: Inputs.range([0., 4], {label: "Temp.", value: 1.0, step: 0.1,}),
    g: Inputs.range([-2, 2], {label: "Tilt", value: 0, step: 0.1}),
    reset: Inputs.button("Reset simulation")
})

// Simulation parameters
dt = 0.005
kB = 1
gamma = 1

// Double well potential: V(x) = x^4 - 2x^2 + g*x
V = (x, g) => x**4 - 2*x**2 + g*x
force = (x, g) => -(4*x**3 - 4*x + g)

// Live simulation with temperature control
simulationState = {
    let x = 0
    let positions = [0]
    let step = 0
    
    while (true) {
        if (params.reset) {
            x = 0
            positions = [0]
            step = 0
        }
        
        let f = force(x, params.g)
        let noise = Math.sqrt(2 * kB * params.T * dt / gamma) * (Math.random() - 0.5) * 2
        x += (f / gamma) * dt + noise
        positions.push(x)
        // Keep only last 500 points for performance
        if (positions.length > 500) {
            positions = positions.slice(-500)
        }
        
        yield {x, positions: [...positions], step: step++}
        await new Promise(resolve => setTimeout(resolve, 10))
    }
}

// Plot
Plot.plot({
    width: 700,
    height: 400,
    grid: true,
    x: {domain: [-2.5, 2.5], label: "Position"},
    y: {domain: [-2, 3], label: "Potential Energy"},
    marks: [
        // Potential curve
        Plot.line(
            Array.from({length: 200}, (_, i) => {
                let x = -2.5 + 5 * i / 199
                return {x, y: V(x, params.g)}
            }),
            {x: "x", y: "y", stroke: "red", strokeWidth: 2}
        ),
        // Particle trajectory (trail)
        Plot.line(
            simulationState.positions.map((x, i) => ({x, y: V(x, params.g), step: i})),
            {x: "x", y: "y", stroke: "blue", strokeWidth: 1, opacity: 0.5}
        ),
        // Current particle position
        Plot.dot([{
            x: simulationState.x, 
            y: V(simulationState.x, params.g)
        }], {x: "x", y: "y", fill: "blue", r: 6, stroke: "white", strokeWidth: 2})
    ]
})

Elementary constituents and energy scales

• Soft matter systems are made of many parts
  
• The assembly of these many parts can be easily deformed.
• Interactions between these parts are weak compared to thermal or mechanical forces.

Coarse graining

Coarse graining

Classes of systems

In our exploration of soft matter we will focus on six main classes of systems which display different physics:

• colloidal dispersions
• polymeric systems
• liquid crystals
• surfactant aggregates
• arrested systems
• active matter

Colloidal dispersions

Polymeric systems

• Polymers are long-chain macromolecules made of repeating monomers.
• Their properties result from a balance of entropy and energy.
• Two main types: synthetic polymers (e.g., plastics) and biopolymers (e.g., DNA, proteins).
• Entanglement: chains cannot cross, leading to unique mechanical behavior.

Polymer entanglement, from Likhtman and Ponmurugan, Macromolecules (2014)

Liquid crystals

• Liquid crystals form when anisotropic soft matter units (e.g., rod-like or disk-shaped molecules) pack densely, leading to partial order—intermediate between liquids and crystals.

• Continuum free energy theories describe liquid crystals by considering the symmetry of their order parameters.

• Liquid crystals are crucial in technologies like liquid crystal displays (LCDs).

Texture of nematic liquid crystals, from https://doi.org/10.3986/alternator.2020.38

Surfactant aggregates

• Surfactants lower surface tension between fluid phases by accumulating at interfaces.
• They are molecules with both hydrophilic heads and hydrophobic tails.
• Surfactants self-assemble at interfaces, forming structures like bilayers and vesicles.
• These assemblies are essential in cell biology and many soft matter systems.

Cellular membrane

Arrested systems

• Arrested systems are trapped in disordered, non-equilibrium states.
• Lack of long-range order leads to slow structural relaxation and prevents reaching the global energy minimum.
• Examples: glasses and gels
• These systems exhibit rigidity, elasticity, and plasticity: disordered (semi)-solids.
• Relaxation times are longer than observable timescales, making equilibrium statistical mechanics insufficient.

An gel (left) and a glass (right) from molecular dynamics with local structure highlighted un colours

Active matter

• Active matter: systems of units that consume energy to move or exert forces.
• Inherently out of equilibrium due to continuous energy input.
• Includes biological systems (bacteria, cells, flocks) and synthetic systems (self-propelled colloids).
• Shows emergent collective behaviors: swarming, clustering, pattern formation.
• Studied using hydrodynamic theories and agent-based models.

Example of active systems, from Cichos et al. Nature Machine Intelligence (2020)

Kinds of colloids

• Colloids = dispersed phase + dispersion medium

Kinds of colloids

IUPAC definition

The International Union of Pure and Applied Chemistry (IUPAC) defines

colloidal: The term refers to a state of subdivision, implying that the molecules or polymolecular particles dispersed in a medium have at least in one direction a dimension roughly between 1 nm and 1 \mum , or that in a system discontinuities are found at distances of that order.

• Why the size?

Colloidal scale

• Particle of size R (micrometric) in medium with constant collisions at thermal energy \sim 1 k_B T
• Compare the energy with the potential energy of settling over a length R
• Obtain a nondimensional number the gravitational Péclet number

\mathrm{Pe}_g=\frac{\Delta m g R}{k_B T}

Destabilisation

• Milk is an emulsion where droplets of fat are dispersed in water and stabilised by proteins.

• If lemon juice is added, the dispersion medium (water) changes, the pH drops, and the emulsion is destabilised.

• The interactions change → separation of curds (solid) and whey (liquid).

Curds and whey resulting from the destabilisation of milk, a colloidal emulsion. (Wikimedia)

Which interactions?

• At the colloidal scale: gravity and electrostatics (eventually magnetostatics).
• Electrostatics forces are the main microscopic forces: colloids are typically charged
• Separation of length and timescales: colloids are much larger and slower than the atomic or molecular constituents of the surrounding fluid.
• Colloid–colloid interactions result from the collective effect (sum or average) of many microscopic interactions, evaluated over times much longer than microscopic relaxation times.
• When we focus on colloidal scales, we integrate out the microscopic degrees of freedom (coarse-graning)
• Coarse-grained interactions are then obtained and are called effective interactions.